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Statistical Inference

Suppose we have a variable
XX
(may or may not be a random variable) that represents the state of nature. We observe a variable
YY
which is obtained by some model of the world
PYXP_{Y|X}
.
Figure 2: Inference Setup
Suppose we know that
XπX\sim \pi
where
π\pi
is a probability distribution. If we observe
Y=yY=y
, then the a posteriori estimate of
XX
is given by Bayes Rule
Pr{X=xY=y}=PYX(yx)π(x)x~PYX(yx~)π(x~)PYX(yx)π(x).\text{Pr}\left\{X=x | Y=y\right\} = \frac{P_{Y|X}(y|x)\pi(x)}{\sum_{\tilde{x}}P_{Y|X}(y|\tilde{x})\pi(\tilde{x})} \propto P_{Y|X}(y|x)\pi(x).
Since the estimate is only dependent on the model and the prior, we don’t actually need to compute the probabilities to figure out the most likely
XX
.

Definition 77

The Maximum A Posteriori (MAP) estimate is given by
X^MAP(y)=argmaxxPYX(yx)π(x)\hat{X}_{MAP}(y) = \text{argmax}_x P_{Y|X}(y|x)\pi(x)
If we have no prior information on
XX
, then we can assume
π\pi
is uniform, reducing Definition 77 to only optimize over the model.

Definition 78

The Maximum Likelihood (ML) estimate is given by
X^ML(y)=argmaxxPYX(yx)\hat{X}_{ML}(y) = \text{argmax}_x P_{Y|X}(y|x)

Binary Hypothesis Testing

Definition 79

A Binary Hypothesis Test is a type of statistical inference where the unknown variable
X{0,1}X\in\{ 0, 1 \}
.
Since there are only two possible values of
XX
in a binary test, there are two “hypotheses” that we have, and we want to accept the more likely one.

Definition 80

The Null Hypothesis
H0H_0
says that
YPYX=0Y\sim P_{Y|X=0}

Definition 81

The Alternate Hypothesis
H1H_1
says that
YPYX=1Y\sim P_{Y|X=1}
With two possible hypotheses, there are two kinds of errors we can make.

Definition 82

A Type I error (false positive) is when we incorrectly reject the null hypothesis. The Type I error probability is then
Pr{X^(Y)=1X=0}\text{Pr}\left\{\hat{X}(Y) = 1 | X = 0\right\}

Definition 83

A Type II error (false negative) is when we incorrectly accept the null hypothesis. The Type II error probability is then
Pr{X^(Y)=0X=1}\text{Pr}\left\{\hat{X}(Y) = 0 | X = 1\right\}
Our goal is to create a decision rule
X^:Y{0,1}\hat{X}: \mathcal{Y} \to \{0, 1\}
that we can use to predict
XX
. Based on what the decision rule is used for, there will be requirements on how large the probability of Type I and Type II errors can be. We can formulate the search for a hypothesis test as an optimization. For some
β[0,1]\beta \in [0, 1]
, we want to find
X^β(Y)=argminPr{X^(Y)=0X=1}:Pr{X^(Y)=1X=0}β.(1)\hat{X}_\beta(Y) = \text{argmin} \text{Pr}\left\{\hat{X}(Y)=0 | X=1\right\} \quad : \quad \text{Pr}\left\{\hat{X}(Y)=1|X=0\right\} \leq \beta. \qquad (1)
Intuitively, our test should depend on
pYX(y1)p_{Y|X}(y|1)
and
pYX(y0)p_{Y|X}(y|0)
since these quantities give us how likely we are to get our observations if we knew the ground truth. We can define a ratio that formally compares these two quantities.

Definition 84

The likelihood ratio is given by
L(y)=pYX(y1)pYX(y0)L(y) = \frac{p_{Y|X}(y|1)}{p_{Y|X}(y|0)}
Notice that we can write MLE as a threshold on the likelihood ratio since if
L(y)1L(y) \geq 1
, then we say
X=1X=1
, and vice versa. However, there is no particular reason that
11
must always be the number at which we threshold our likelihood ratio, and so we can generalize this idea to form different forms of tests.

Definition 85

For some threshold
cc
and randomization probability
γ\gamma
, a threshold test is of the form
X^(y)={1 if L(y)>c0 if L(y)<c Bernoulli(γ) if L(y)=c.\hat{X}(y) = \begin{cases} 1 & \text{ if } L(y) > c\\ 0 & \text{ if } L(y) < c\\ \text{ Bernoulli}(\gamma) & \text { if } L(y) = c. \end{cases}
MAP fits into the framework of a threshold test since we can write
X^MAP={1 if L(y)π0π10 if L(y)<π0π1\hat{X}_{MAP} = \begin{cases} 1 & \text{ if } L(y) \geq \frac{\pi_0}{\pi_1}\\ 0 & \text{ if } L(y) < \frac{\pi_0}{\pi_1} \end{cases}
It turns out that threshold tests are optimal with respect to solving Equation 1.

Theorem 44 (Neyman Pearson Lemma)

Given
β[0,1]\beta\in[0, 1]
, the optimal decision rule to
X^β(Y)=argminPr{X^(Y)=0X=1}:Pr{X^(Y)=1X=0}β\hat{X}_\beta(Y) = \text{argmin} \text{Pr}\left\{\hat{X}(Y)=0 | X=1\right\} \quad : \quad \text{Pr}\left\{\hat{X}(Y)=1|X=0\right\} \leq \beta
is a threshold test.
When
L(y)L(y)
is monotonically increasing or decreasing, we can make the decision rule even simpler since it can be turned into a threshold on
yy
. For example, if
L(y)L(y)
is monotonically inreasing, then an optimal decision rule might be
X^(y)={1 if y>c0 if y<cBernoulli(γ) if y=c.\hat{X}(y) = \begin{cases} 1 & \text{ if } y > c\\ 0 & \text{ if } y < c\\ \text{Bernoulli}(\gamma) & \text{ if } y = c. \end{cases}